Multi-segment and nonlinear droop control for parallel operating active front end power converters

ABSTRACT

An active rectifier includes first and second DC nodes, a switching circuit, and a controller configured to compute a voltage reference according to a load signal of the DC output, and a non-linear relationship between a load condition of the DC output and a DC bus voltage at the DC output, and to generate rectifier switching control signals according to the voltage reference to cause the switching circuit to convert AC input power from the AC input to control the DC bus voltage at the DC output.

BACKGROUND INFORMATION

The subject matter disclosed herein relates to power conversion systemsand active rectifiers.

BRIEF DESCRIPTION

Various aspects of the present disclosure are now summarized tofacilitate a basic understanding of the disclosure, wherein this summaryis not an extensive overview of the disclosure, and is intended neitherto identify certain elements of the disclosure, nor to delineate thescope thereof. Rather, the primary purpose of this summary is to presentthe concept of the disclosure in a simplified form prior to the moredetailed description that is presented hereinafter.

An active rectifier is disclosed, having first and second DC nodes, aswitching circuit, and a controller configured to compute a voltagereference according to a load signal of the DC output, and a non-linearrelationship between a load condition of the DC output and a DC busvoltage at the DC output, and to generate rectifier switching controlsignals according to the voltage reference to cause the switchingcircuit to convert AC input power from the AC input to control the DCbus voltage at the DC output.

A power conversion system includes a first active rectifier with firstand second DC nodes, a first switching circuit, and a first controllerconfigured to compute a first voltage reference according to a firstload signal of a first DC output, and a first non-linear relationshipbetween a first load condition of the first DC output and a DC busvoltage at the DC output. The first controller is further configured togenerate first rectifier switching control signals according to thefirst voltage reference to cause the switching circuit to convert firstAC input power from the first AC input to control the DC bus voltage.The system includes a second active rectifier with a second switchingcircuit, and a second controller configured to compute a second voltagereference according to a second load signal of a second DC output, and asecond non-linear relationship between a second load condition of thesecond DC output and a DC bus voltage at the DC output. The secondcontroller is further configured to generate second rectifier switchingcontrol signals according to the second voltage reference to cause thesecond switching circuit to convert second AC input power from thesecond AC input to control the DC bus voltage.

A method includes computing a voltage reference according to a loadsignal of a DC output of an active rectifier, and a non-linearrelationship between a load condition of the DC output and a DC busvoltage at the DC output, as well as generating switching controlsignals for the active rectifier according to the voltage reference tocause a switching circuit of the active rectifier to convert AC inputpower from an AC input to control the DC bus voltage.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram.

FIG. 2 is a flow diagram.

FIG. 3 is a graph.

FIG. 4 is a graph.

FIG. 5 is a waveform diagram.

FIG. 6 is a waveform diagram.

FIG. 7 is a waveform diagram.

FIG. 8 is a waveform diagram.

FIG. 9 is a waveform diagram.

FIG. 10 is a waveform diagram.

FIG. 11 is a waveform diagram.

FIG. 12 is a waveform diagram.

DETAILED DESCRIPTION

Referring now to the figures, several embodiments or implementations arehereinafter described in conjunction with the drawings, wherein likereference numerals are used to refer to like elements throughout, andwherein the various features are not necessarily drawn to scale. In thefollowing discussion and in the claims, the terms “including”,“includes”, “having”, “has”, “with”, or variants thereof are intended tobe inclusive in a manner similar to the term “comprising”, and thusshould be interpreted to mean “including, but not limited to . . . ”Also, the term “couple” or “couples” is intended to include indirect ordirect electrical or mechanical connection or combinations thereof. Forexample, if a first device couples to or is coupled with a seconddevice, that connection may be through a direct electrical connection,or through an indirect electrical connection via one or more interveningdevices and connections.

Active front end (AFE) rectifiers (e.g., also referred to as AFEconverters) can regenerate power to an AC grid or convert AC input powerto a DC bus. Unlike passive (e.g., diode) rectifiers, AFE rectifiers canprovide power factor control (PFC), for example, to provide at or nearunity power factor, with minimum current distortion (e.g., typicallyless than 5% according to IEEE standards). The AFE rectifier can be usedin motor-drive applications where the AFE DC output is connected tomultiple inverters or to one inverter controlling motor speed and/ortorque. The AFE rectifier can also be used in grid tie applicationswhere the DC side is connected to a DC Bus supply such as a fuel cell,or solar cell, etc. Droop control can be used for controlling multipleAFE rectifiers connected to a shared DC bus. One droop control approachinvolves detecting the angle difference between rectifiers and uses realtime communication between all units. Other approaches use rectifierswitching frequency control.

FIG. 1 shows a power conversion system 100 with a DC bus shared bymultiple components. The system 100 includes an AC grid 102, such as athree-phase grid, although other examples are possible using a singlephase grid or a multi-phase grid having more than three phases (notshown). A transformer 104 is connected between the three phase grid 102and multiple multiphase AC connections. In one example, a primarywinding of the transformer 104 is connected to the AC grid 102, and thetransformer 104 includes multiple secondary windings (not shown). Inthis example, a set of three first AC nodes 106 are connected to a firstsecondary of the transformer 104, and a set of three second AC nodes 108are connected to a second secondary of the transformer 104. In otherexamples, one or more of the secondaries can be single phasearrangements, or multiphase arrangements having more than three phases(not shown).

The example of FIG. 1 includes first and second three-phase secondariesconnected to respective first and second active rectifiers 111 (labeledAFE1) and 112 (labeled AFE2). In other implementations, more than twoactive rectifiers can be connected to a single shared DC bus. The firstrectifier 111 includes a first AC input, including the first AC nodes106, as well as a first DC output. The first rectifier 111 also includesa first controller. In one example, the controller 113 includes aprocessor 114 (e.g., a microprocessor, microcontroller, FPGA, logiccircuit, etc.). The processor 114 is operatively connected to anelectronic memory 115 that stores program instructions and/or data. Inone example, the memory 115 stores program instructions that implement adroop control component 116 as described further hereinafter. The firstrectifier 111 also includes a first switching circuit 118 coupled withthe first AC input and with the first DC output. The second rectifier112 is similar to the first rectifier 111, and includes a second ACinput, including the second AC nodes 108, a second DC output connectedto the first DC output of the first rectifier 111 along the shared DCbus. The second rectifier 112 also includes a controller 113 with aprocessor 114, and an electronic memory 115 that stores programinstructions and/or data to implement a droop control component 116.

FIG. 1 also shows a shared DC bus 120 with first and second DC nodes 121(labeled DC+) and 122 (labeled DC−), respectively, connected to the DCoutputs of the rectifiers 111 and 112. In motoring operation, therectifiers 111 and 112 individually convert corresponding AC input powerto develop a DC bus voltage VDC between the first and second DC nodes121 and 122. In particular, the first active rectifier 111 provides afirst DC output current signal I_(PU1) to the DC bus 120, and the secondactive rectifier 112 separately provides a second DC output currentsignal I_(PU2) to the DC bus 120, where the “PU” designation indicatesper-unit valuation. The example power conversion system 100 includes aswitching inverter 124 with a DC input connected to the DC bus 120, aswell as a photovoltaic (PV) system 126 with a DC output connected to theDC bus 120. In operation, the first and second rectifiers 111 and 112can each deliver power from the respective AC nodes 106, 108 to the DCbus 120 in “motoring” operation, and can transfer power from the DC bus120 to the respective AC nodes 106, 108 in “regeneration” or “regen”operation, for example, where a motor load (not shown) driven by theinverter 124 is decelerating, or where energy from the photovoltaicsystem 126 is being delivered through the DC bus 120 and the inverters111, 112 to the grid 102.

FIG. 1 shows further details of one of the switching circuits 118,including switching devices S1-S6 individually connected between acorresponding one of the respective AC input lines 106, 108 and acorresponding node 121 or 122 of the shared DC bus. The respectiveswitching devices S1-S6 are configured to selectively couple one of theAC nodes 106 or 108 with a respective one of the first and second DCnodes 121, 122 according to a respective switching control signal. Theswitching circuit 118 in one example includes a driver circuit 130 thatprovides switching control signals 132 to the respective switchingdevices S1-S6 under control of the processor 114 of the controller 113.The switching devices S1-S6 are respectively configured to selectivelycouple a corresponding one of the first AC nodes 106 with a respectiveone of the first and second DC nodes 121, 122 according to a respectiveswitching control signal 132 from the controller 113 (e.g., via anyintervening driver circuit 130) for motoring and/or regenerationoperation of the corresponding AFE rectifier 111, 112.

FIG. 1 shows further details of an example storage organization in therespective AFE controller memories for processor-implementation of thedroop control component 116 in the electronic memory 115. The exampledroop control components 116 of the respective rectifiers 111 and 112operate independently of one another. In this manner, the rectifiers 111and 112 do not need to communicate with one another for shared controlof the DC bus voltage VDC of the DC bus 210. Moreover, the droop controlcomponents 116 implement intelligent non-linear and/or multi-segmentdroop control to mitigate rectifier derating. Furthermore, the exampledroop control components 116 of the respective rectifiers 111 and 112can be configured similarly or can have different configurationparameters and settings, for instance, to facilitate load sharingbetween rectifiers of different sizes or ratings, to operate the sharedDC bus 120.

The example droop control component 116 in the rectifier controllers 113include program code or instructions 140 that cause the processor 114 toimplement the droop control concepts described herein. The droop controlcomponent 116 in one example includes an integer number “n” linearequations and associated parameters 142, 144, where n is greater than orequal to 2. The parameters 142, 144 define a multi-segment function withtwo or more segments having different respective slopes that relate aload condition of the DC output to the DC bus voltage VDC. In anotherimplementation, the droop control component 116 includes one or moresecond or higher order formulas and associated equation parameters 146that relate the respective load conditions of the DC output to the DCbus voltage VDC. In one example, the droop control component 116 in therectifier controller 113 includes a lookup table 148 with entries thatrelate load conditions of the DC output to respective DC bus voltagesVDC at the DC output according to the second or higher order formula.

In one example, the droop control component 116 in the rectifiercontroller 113 also includes (e.g., samples and stores in the memory115) various operating parameters or values and computed referencevalues 150 used in operation of the rectifier 111, 112. On exampleincludes a root-mean-square line to line voltage VLL_(RMS) of the ACinput for a given control cycle, a load signal (e.g., DC output currentsignal I_(PU)) of the DC output, a sampled DC bus voltage feedbacksignal VDC_(FB), a computed minimum DC bus voltage value VDC_(MIN)(t)for a given rectifier switching control cycle, a computed no load DC busvoltage value VDC₀(t) for the given control cycle, a constant k, and acomputed voltage reference VDC_(REF) for the given control cycle. Theroot-mean-square line to line voltage VLL_(RMS), the load signal (e.g.,DC output current signal I_(PU)) of the DC output, and the sampled DCbus voltage feedback signal VDC_(FB) for each rectifier 111, 112 aresampled and provided to the respective controllers 113 for each controlcycle in one example.

Referring also to FIG. 2, the controller 113 of FIG. 1 in one exampleimplements the droop control component 116 according to a method 200 inFIG. 2. The method 200 in one example is implemented concurrently inmultiple active rectifiers (e.g., rectifiers 111 and 112 in FIG. 1) forindependent autonomous operation of the parallel units four balancedload sharing with droop control. FIG. 2 illustrates operation of asingle one of the active rectifiers 111, 112, and the other rectifier issimilarly configured and operated.

At 202, the controllers 113 are configured with parameters formulti-segment and/or non-linear formulas and/or lookup table entries toimplement multiple autonomous rectifier operation in the shared DC bussystem 100, such as the elements and entries 142, 144, 146 and/or 148 inthe respective droop control components 116 of FIG. 1. The configurationdata in one example is stored in the electronic memory 115 at 204. Inone example, the controller 113 is configured by an externalconfiguration source, such as a computer that sends configurationinformation to the controller 113 by a network connection (not shown).In another example, the rectifier 111, 112 includes a user interface(not shown), and the configuration at 202 is performed via theinterface. In one example, the controller 113 is configured at 202 withmulti-segment load thresholds (e.g., endpoints (e.g., segment range andpoint current values I_(EP))) and segment slopes (e.g., a1, a2, . . . )for multi-segment droop control. In another example, the controller 113is configured at 204 with non-linear equation parameters (a, b, c, d) orlookup table entries for droop control.

Each new rectifier switching control cycle begins at 204 in FIG. 2, andthe controller 113 samples input conditions and feedback values at 206.In one example, the controller 113 samples the line-line AC inputvoltage (VLL_(RMS)), the AFE output current feedback (I_(PU)) and the DCbus voltage feedback (VDC_(FB)) at 204 for the present control cycle.For the given control cycle, the controller 113 computes a voltagereference value VDC_(REF) at 210 according to a load signal, such as arectifier output current value I_(PU) of the DC output, and according toa first non-linear relationship between a load condition of the DCoutput and the DC bus voltage VDC across the first and second DC nodes121, 122. At 220 in FIG. 2, the controller 113 generates the switchingcontrol signals 132 according to the computed voltage referenceVDC_(REF) for the given control cycle to cause the switching circuit 118to convert AC input power from the AC input to control the DC busvoltage VDC. As previously discussed, the individual active rectifiers111, 112 each implement the processing in FIG. 2 autonomously, and maybe operated at different switching frequencies with asynchronousswitching cycles, although not a requirement of all possibleimplementations. In the illustrated example of FIG. 2, with first andsecond rectifiers 111 and 112, the first load signal I_(PU1) of thefirst DC output is the first DC output current signal I_(PU1) of thefirst DC output, and the second load signal I_(PU2) of the second DCoutput is the second DC output current signal I_(PU2) of the second DCoutput.

In one example, the non-linear relationship is a multi-segment functionwith multiple segments having different respective slopes that relatethe load condition of the DC output to the DC bus voltage VDC at the DCoutput. In one example, the non-linear relationship is a second orhigher order formula 146 that relates the load condition of the DCoutput to the DC bus voltage VDC at the DC output. In one example, theload signal I_(PU) of the DC output is a DC output current signal I_(PU)of the DC output (e.g., I_(PU1) or I_(PU2)).

In one implementation (e.g., FIG. 3 below), the respective first andsecond non-linear relationships are multi-segment functions withmultiple segments having different respective slopes that relate theload condition of the DC output to the DC bus voltage VDC. In thisexample, the controller 113 computes the DC voltage reference valueVDC_(REF) at 212 for a dual segment implementation according to one oftwo possible ranges of the loading of the rectifier 111, 112. In thisexample, each rectifier 111, 112 is configured with parametersassociated with to power ranges (e.g., ranges of the corresponding loadsignal I_(PU)). In this example, the controller 113 computes the DCvoltage reference value VDC_(REF) for the first range as(−a1*I_(PU))+VDC₀, where a1 is a corresponding first slope used for thefirst range. For a second range, the controller 113 computes the DCvoltage reference value VDC_(REF)=(−a2*I_(PU))+VDC_(EPI), where a2 is adifferent second slope, and VDC_(EP)1 is the endpoint of the definedfirst range, which is the same as the start point of the second range.Any combination of multiple segments can be used in variousimplementations, including multiple linear segments (e.g., piecewiselinear), multiple curvilinear segments, or combinations of at least onelinear segment and at least one curvilinear segment. In the illustratedexample, the multi-segment function includes two or more segments havingdifferent respective slopes that relate the load condition of the DCoutput to the DC bus voltage VDC at the DC output. In another example,more than two segments are used, and at least two of the segments havedifferent respective slopes that relate the load condition of the DCoutput to the DC bus voltage VDC at the DC output.

In another example (e.g., FIG. 4 below), the non-linear relationshipsimplemented by the respective controllers 113 are second or higher orderformulas 146 that relate the load conditions of the respective first andsecond DC outputs to the DC bus voltage VDC. In the example of FIG. 2,the controller 113 implements the computation at 210 by computing aminimum DC bus voltage value at 214 according to the sampled line-lineAC input voltage, computes a no-load DC bus voltage value at 216according to the computed minimum DC bus voltage value, and computes thevoltage reference value at 218 by solving the second or higher orderformula 146, or by indexing a lookup table 148.

In one implementation, the controller 113 selectively adjusts thenon-linear relationship according to the operating condition of therectifier 111, 112, for example, to accommodate changes in the minimumDC bus voltage. In the example of FIG. 2, the controller 113 samples theroot-mean-square (RMS) line to line voltage VLL_(RMS) of the AC inputfor a given control cycle at 206, and uses this in updating or adjustingthe non-linear relationship of the second or higher order formula 146.In this example, the controller 113 computes a minimum DC bus voltagevalue VDC_(MIN)(t) at 214 according to the RMS line to line voltageVLL_(RMS) for the given control cycle. In one implementation, thecontroller 113 computes the minimum DC bus voltage value VDC_(MIN) (t)at 214 as 1.02*VLL_(RMS)*√2. At 216, the controller 113 computes the noload DC bus voltage value VDC₀(t) according to the minimum DC busvoltage value VDC_(MIN)(t) for the given control cycle according to aconstant k as VDC₀(t)=k*VDC_(MIN)(t). At 218, the controller 113computes the voltage reference VDC_(REF) for the given control cycleaccording to the load signal I_(PU), the non-linear relationship, andthe no load DC bus voltage value VDC₀(t). In one example, the controller113 computes VDC_(REF) at 218 using a third order equation asa(I_(PU))3+b(I_(PU))²+c(I_(PU))+VDC₀(t), using the non-linear equationparameters 146 (e.g., a, b, c) configured at 202. In other examples, adifferent second or higher order formula can be used, such as a secondorder equation or a fourth or higher order equation (not shown).

In another implementation, the droop control component 116 implements alookup table 148 that includes entries that relate load condition of theDC output to the DC bus voltage VDC at the DC output according to thesecond or higher order formula to compute the DC voltage reference valueVDC_(REF) at 218. In one implementation, the lookup table is fixed, andthe selective adjustment of the no-load DC bus voltage value is omitted(e.g., 214 and 216 in FIG. 2 can be omitted). In another example, thelookup table includes different sets of entries for different no-load DCbus voltage values, and the DC bus voltage reference is computed at 218by indexing according to a computer no-load DC bus voltage value for thecurrent rectifier control cycle. In one implementation, the controller113 indexes the lookup table 148 at 218 according to the DC outputcurrent value (I_(PU)) (e.g., and optionally according to an updatedcomputed no-load DC bus voltage value) and determines a corresponding DCvoltage reference value VDC_(REF). In one implementation, the controller113 uses linear interpolation or other suitable technique at 210 toselect and/or compute a suitable value of the DC voltage reference valueVDC_(REF) according to the DC output current value I_(PU) based on theentries of the lookup table 148. In one example,

At 220, the controller 113 generates the switching control signals 132for the active rectifier 111, 112 according to the voltage referenceVDC_(REF) to cause a switching circuit to convert the corresponding ACinput power from the AC input to control the DC bus voltage VDC.

Referring also to FIG. 3, each segment of the example multi-segmentlinear droop mechanism (212 in FIG. 2) follows a linear equationy=−ax+b, where “a” is the slope of the line and very critical for loadsharing. VDC_(MIN) is the minimum DC bus operating point (e.g., computedin one example as VLL_(RMS)*sqrt(2)*1.02. VDC₀ is the DC bus at zeroloading condition. VDC is the DC bus voltage at a certain loadingcondition, VDC<VDC₀. P_(T)=P1+P2. VDCmax is the maximum DC bus operatingpoint at full regen condition. VDC is the DC bus voltage at a certainloading condition, VDC>VDC₀. P_(T)=P1+P2. The DC bus can vary fromVDCmax to VDCmin depending on the loading conditions.

A graph 300 in FIG. 3 shows a first curve 301 with segments 301-1, 301-2and 301-3 implemented by the first controller 113 of the first rectifier111, and a second curve 302 and includes segments 302-1, 302-2 and 302-3implemented by the second controller 113 of the second rectifier 112 formulti-segment droop control (e.g., 212 in FIG. 2). Each of the curves301 and 302 is represented on both sides of a zero power axis 310, wherethe second and third illustrated portions of each curve (e.g., 301-2 and301-3, as well as 302-2 and 302-3) are at the same or similar slopes,which are greater than the slopes of the first portions 301-1 and 302-1.The slope settings can be the same for each rectifier 311, 312, or theycan be different for each rectifier 311, 312. In the illustratedexample, the first portion 301-1 of the first curve 301 has a range301-1R, and the first portion 302-1 of the second curve 302 has adifferent range 302-1R. In other implementations, the ranges 301-1R and302-1R can be the same. The graph 300 also shows the no-load DC busvoltage 304 and the minimum DC bus voltage 306 Each segment of eachmulti-segment curve has at least one and point. The graph 300 showsoperation at one example total power operating point for motoringoperation within a motoring range 312, where the DC bus voltage VDC isshown as a point 308. For regeneration operation, the graph 300 shows aregeneration region 314 above the no-load DC bus voltage 304. At theillustrated motoring operating condition, the curve 301 provides foroperation of the first rectifier 111 in a first power setting P1,designated 311 in FIG. 3, and the second curve 302 provides foroperation of the second rectifier 112 and a second power setting P2,designated 312, where the total power P_(T)=P1+P2.

The droop gain for each segment of each rectifier curve in FIG. 3represents the slope a in the equation y=−ax+b. In the AFE bus supplyparallel application, y is the VDC_(REF) (e.g., the DC bus voltage at acertain loading condition), x is the current in per unit (e.g., I_(PU)),and b is the DC bus voltage at no load condition VDC₀ for the firstsegments 301-1, 302-1. Different intercepts for the other segments aredetermined according to the endpoint of the adjacent segments.

In one example where the droop gain=5%, 480 volt, at no load conditionthe operating voltage VDC=VDC₀=1.05*VDC_(MIN). Moreover, at full loadregen condition the operating voltage VDC=1.1*VDC_(MIN). Therefore, thevoltage boost needed is 0.1*480*sqrt(2)*1.02=70 volts. Increasing the DCbus voltage VDC results in increased system losses for single segmentlinear droop control (IGBT, LCL inductor and capacitors) and systemderating that can reach up to 35% of the drive rating.

The droop control component 116 of the controller 113, and the method200 can be implemented to alleviate this issue and provide improvedpower operation under several operating conditions. In the exampledual-segment implementation, the droop mechanism for each of therectifiers 111, 112, illustrated by the example curves 301 and 302, canbe represented by two linear equations y=−a1x+b1&y=−a2x+b2. At lightload condition the droop (e.g., the segment slope al of the firstsegment 301-1) can be chosen to have a small value, such as “1% or 2%”since accurate load balancing is not important at light load conditions.At higher load conditions, the droop (e.g., the slope of the furthersegments 301-2 and 301-3) is chosen to have higher values, such as “4%or 5%”. The effective DC bus voltage boost will be reduced whilemaintaining a very good load sharing at higher loading conditions.Assuming the minimum DC bus reference based on the input line voltage is692 volts, for a standard droop (e.g., 4%), at no load, the DC busvoltage reference VDC_(REF)=692+692*0.04=719.7 V. At full load regen,VDC_(REF)=692+692*0.08=747.4 V. Derating=−0.528*delta−Vbus=29% at fullregen. For a two segment droop (e.g., 2% from 0 to 0.5 pu and 4% from0.5 pu to 1 pu), the average droop is 3%. At no loadVDC_(REF)=692+692*0.03 =712.5 V. At full load regen,VDC_(REF)=692+692*0.06=733.5 V. Operating the system at a lower DC busvoltage results in efficient operation and reduce stress on the drivecomponents As previously discussed, the multi-segment droop controltechnique can be extended to three-segment droop, fourth-segment droop,where the general case will be the nonlinear droop.

FIG. 4 includes a graph 400 that shows implementation of non-lineardroop control (e.g., 218 in FIG. 2 above). The graph shows a first curve401 for the first rectifier 111, and a second curve 402 for the secondrectifier 112, including a zero power axis 310, the no-load DC busvoltage 304, the minimum DC bus voltage 306, and both a motoring region312 and a regeneration region 314 as previously described. In thisexample, the operating DC bus voltage 308 corresponds to a first powerP1 (311) for the first rectifier 111, and a second power P2 (312) forthe second rectifier 112. In this example, the droop mechanism isrepresented by one nonlinear equation y=ax³+bx²+cx+d, where “y” is theDC bus voltage and “x” is the loading in per unit (e.g., I_(PU)). Theslope of the curve varies based on the operating point dy/dx=3ax²+2bx+c.The slopes (e.g., droop gain) can be determined at three operating pointand the coefficients can be calculated in real time based on the minimumDC bus voltage “VDC_(MIN)”. In this example, a third order curve isused, although a second order formula can be used in otherimplementations, and further examples can use higher order formulas.

The slope “droop gain” can be calculated at dy/dx=y^(\)=3ax²+2bx+c. Forexample, designating the droop at 10% loading to be D1, the droop at 50%loading to be D2, and 75% loading to be D3, the following equationsapply:

y ^(\)(0.1)=−0.01*VDC_(MIN);  (1)

y ^(\)(0.5)=−0.02*VDC_(MIN);  (2)

y ^(\)(0.75)=−0.04*VDC_(MIN);  (3)

At full load condition, the DC bus voltage will not go below VDC_(MIN),therefore

a+b+c+d=VDC_(MIN)  (4)

In one example, the minimum DC bus voltage VDC_(MIN) is a predeterminedvalue, which can be calculated as VLL_(RMS)*sqrt(2)*1.02 (e.g., at 214in FIG. 2). As VDC_(MIN) may change in real time based on the calculatedroot mean square value of the input line voltage, the controller 113 inone example solves equations 1-4 in real time for each given rectifierswitching control cycle in order to adjust the curve according toVDC_(MIN).

The following illustrates computations for an example channel-channeldroop control implementation using multi-segment droop control (e.g.,212 in FIG. 2):

V=V ₀ −KP  (5)

Set V₀=1P. u, K=0.04

at full load condition:

V=1−(0.04)(1)=0.93 p. u.  (5a)

the following applies for a two segment droop implementation:

V=V₀ ¹−K₀P₀−K₁(P−P₀)

Where P>P₀

Therefore, V=V ₀ ¹ −K ₀ P ₀ −K ₁(P−P ₀)  (6)

Comparing equations 5 and 6 yields the following:

V₀=V₀ ¹−K₀P₀+K₁P₀

set K₀=0.02, P₀=0.5, K₁=0.04

Therefore 1=V₀ ¹−(0.02)(0.5)+(0.04)(0.5)

Therefore V₀ ¹=0.99 p. u.

At full load conditions, the following applies:

V=0.99−(0.02)(0.5)+(0.04)(0.5)−(0.04)  (5)

0.99−0.01+0.02−0.04  (6a)

In this example, the final voltage from (5a) and (6a) is the same.

In one example, for 1 P.U.=720 volts, V₀=720

V₀ ¹=(720)(0.99)=712.8

In both cases “final voltage V after droop” is given by:

V=(0.96)(720)=691.2

The following illustrates computations for an example channel-channeldroop control implementation using nonlinear droop control (e.g., 218 inFIG. 2):

y=ax ³ bx ² +cx+d  (7)

y=DC bus voltage

x=loading in p.u.

therefore y¹=3ax²+2bx+c

If the non-linear curve is designed such that droop=0.01 at 10% loading,droop=0.02 at 50% loading, and droop=0.04 at 75% loading, the followingapplies:

y¹(0.1)=(−0.01)(VDC_(min))

y¹(0.5)=(−0.02)(VDC_(min))

y¹(0.75)=(−0.04)(VDC_(min))

Where VDC_(min)=(V_(LL))(√{square root over (2)})(1.02)

therefore 3a(0.1)²+2b(0.1)+c=(−0.01)(VDC_(min))

Also,

3a(0.5)²+2b(0.5)+C=(−0.02)(VDC_(min))

Also,

3a(0.75)²+2b(0.75)+C=(−0.04)(VDC_(min))

therefore

0.03a+0.2b+c=(−0.01)(VDC_(min))  (7)

0.75a+b+c=(−0.02)(VDC_(min))  (8)

1.6875a+1.5b+c=(−0.04)(VDC_(min))  (9)

And at full load the DC Bus voltage cannot go below VDC_(min)

Therefore a+b+c+d=VDC_(min)  (10)

Assembled, (1), (2), (3), (10) provides the following:

${\begin{bmatrix}{{0.0}3} & {0.2} & 1 & 0 \\{{0.7}5} & 1 & 1 & 0 \\{{1.3}875} & 1.5 & 1 & 0 \\1 & 1 & 1 & 0\end{bmatrix}\begin{bmatrix}a \\b \\c \\d\end{bmatrix}} = {\begin{bmatrix}{{- {0.0}}1} \\{{- {0.0}}2} \\{{- {0.0}}4} \\1\end{bmatrix}{VD}C_{\min}}$

Solving this set of equations for (9) yields the following:

V_(LL)=480+√{right arrow over (2)}+1.02=692.3

a=−19.529, b=8.9213

c=−8.1224, d=711.1303

therefore y=(−19.529)(x³)+(8.9213)(x²)+(−8.1224)(x)+711.1303

-   -   at x=0, y=VDC=711.1303    -   at x=1    -   y¹=3(−19.529)(1)²+2(8.9213)(1)+(−8.1224)    -   y¹=−48.86    -   therefore

${droop} = {\frac{{- 4}{8.8}6}{692.3} = {{- {0.0}}7}}$

-   -   notice that a, b, c, d has to be solved in real time as Vd_(min)        varies in real time.    -   For regen operation, curve (10)    -   Odd symmetry can be used    -   where ΔVd_(motor)=VDC₀−(ax³+bx²+cx+d)    -   therefore VDC_(regen)=VDC₀+ΔVDC_(motor)    -   notice ΔVDC_(regen) at the same loading condition for 480 volts    -   VDC_(max)=711+(711−692.3)=729.7

The following is an example nonlinear method of finding VDC_(REF)reference based on nonlinear equation y=ax³+bx²+cx+d. The method 200above in one example controls droop indirectly in a parabolic shapebased on cubic VDC_(REF) equation. In one example, setting the no loaddroop gain to be D₁, the full load droop gain to be D₂, and a droopboost to be D₃, a no load droop gain percent (D_(nl))=D₁*(1−D₃), and afull load droop gain percent (D_(fl))=D₂*(1+D₃). In this example, thefollowing computations apply:

V _(DCDroopRef)=−((⅓)(D _(fl) −D _(nl)))(I _(qREF))³ −D _(nl)(I_(qREF))+(⅓)(D _(fl)+2*D _(nl))  (11)

V _(DCRef) =V _(DCDroopRef) +V _(DCMinOpt) (units in pu)  (12)

V _(DCDroop)=−(D _(fl) −D _(nl))(I _(qREF))² −D _(nl)   (13)

Parameter limits for this example are given as follows:

V _(DCMaxPu)=(V _(DCMax) /V _(DCMinOpt))−1  (14)

D _(nlMax)=Min(0.05, D _(nl))  (15)

Using Full load regen I_(gREP=(−)1) pu in equation (1) at max V_(DCRef),

D_(flMax)=(½)(3*V_(DCMaxPu)−4*D_(nl))

Referring to FIGS. 5 and 6, FIG. 5 shows a graph 500 with a curve 502showing a linear DC bus voltage reference VDC_(REF) as a function ofloading for motoring operation, and a curve 504 showing a cubicnon-linear DC bus voltage reference VDC_(REF). FIG. 6 shows a graph 600with a curve 602 of linear droop curve as a function of loading formotoring operation and regen, and a curve 604 showing an example cubicnon-linear droop control formula. In this example, the inputs are asfollows:

Inputs: I_(qREF)=−1 to 1 pu in 2 secs

D_(nl)=0.01

D_(fl)=0.09

Droop Boost=0.2 (20%)

Referring to FIGS. 7 and 8, FIG. 7 shows a graph 700 with a curve 702showing a linear DC bus voltage reference VDC_(REF) as a function ofloading for motoring operation, and a curve 704 showing a cubicnon-linear DC bus voltage reference VDC_(REF). FIG. 8 shows a graph 800with a curve 802 of linear droop curve as a function of loading formotoring operation, and a curve 804 showing an example cubic non-lineardroop control formula. In this example, the inputs are as follows:

Inputs: I_(qREF)=−1 to 1 pu in 2 secs

D_(nl)=0.0

D_(fl)=0.05

Droop Boost=0.2 (20%)

Referring to FIGS. 9 and 10, FIG. 9 shows a graph 900 with a curve 902showing a linear DC bus voltage reference VDC_(REF) as a function ofloading for motoring operation, and a curve 904 showing a cubicnon-linear DC bus voltage reference VDC_(REF). FIG. 10 shows a graph1000 with a curve 1002 of linear droop curve as a function of loadingfor motoring operation, and a curve 1004 showing an example cubicnon-linear droop control formula. In this example, the inputs are asfollows:

Inputs: I_(gREF)=−1 to 1 pu in 2 secs

D_(nl)=0.01

D_(fl)=0.08

Droop Boost=0.2 (20%)

Referring to FIGS. 11 and 12, FIG. 11 shows a graph 1100 with a curve1102 showing a linear DC bus voltage reference VDC_(REF) as a functionof loading for motoring operation, and a curve 1104 showing a cubicnon-linear DC bus voltage reference VDC_(REF). FIG. 12 shows a graph1200 with a curve 1202 of linear droop curve as a function of loadingfor motoring operation, and a curve 1204 showing an example cubicnon-linear droop control formula. In this example, the inputs are asfollows:

Inputs: I_(qREF)=−1 to 1 pu in 2 secs

D_(nl)=0.01

D_(fl)=0.12

Droop Boost=0.2 (20%)

V_(DCmax)=770V

V_(DCmin)=700V

Another aspect provides non-linear droop control by defining the desiredoutput droop gain as a second order curve relationship between the droopgain and the active q-axis current as below:

droopGain=a×l _(q) ² +b×l _(q) +c  (15)

Where the coefficients a, b, and c are calculated depending on two droopgain parameters (k1=light load gain, and k2=high load gain) and a q-axistransition current (Iq_Trans) which can form any second order droopcurve in equation (15).

By integrating the second order Droop Gain equation (15), the DC linkdroop voltage reference can be calculated as follows:

$\begin{matrix}{{V_{{{dc}\_ {droop}}{\_ {ref}}} = {{\frac{a}{3} \times I_{q}^{3}} + {\frac{b}{2} \times I_{q}^{2}} + {c \times I_{q}} + c_{1}}}{{{Where}\mspace{14mu} c_{1}} = {- \left( {\frac{a}{3} + \frac{b}{2} + c} \right)}}} & (16)\end{matrix}$

represents the No-load Droop DC link voltage at Iq=0.

In various implementations, a linear single slope droop is fairly simpleto implement, but suffers from significant DC bus voltage increase thatreduces the efficiency of the drive and cause severe de-rating. Theexample dual segment droop (e.g., FIG. 3 above, where the second andthird segments are identical, but separately used for motoring andregeneration) uses two different slopes. As a result, the DC bus voltageincrease is less than the standard droop, which can facilitate improvedefficiency and reduced de-rating. The concept can be generalized asdescribed above to an integer number n droop segments, where n isgreater than or equal to 2. The nonlinear droop (e.g., FIG. 4 above)provides one or more non-linear droop curves in the form ofy=a₁xn+a₂x^((n−1))+a₃x^((n−2))+ . . . +a_(n). In spite of potentiallyincreased complexity, the nonlinear droop results in good performance byreducing the DC bus voltage boost while maintain a sharing unbalance of5% at full load condition in one example. The reduction of DC busvoltage boost reduces inductor losses in any included LCL filter, andcan help reduce the deratings of the overall system. Various possibleimplementations can increase the effective drive rating, such as formotor drive applications, and increase the system efficiency whilemaintaining adequate load sharing between the units working in parallel.Moreover, specific implementations can combine multiple activerectifiers of different ratings, with the autonomous multi-segmentand/or non-linear droop control providing the above advantages inaddition to flexibility of pairing multiple active rectifiers in ashared DC bus system.

In the preceding specification, various embodiments have been describedwith reference to the accompanying drawings. It will be evident thatvarious modifications and changes may be made thereto, and additionalembodiments may be implemented, without departing from the broader scopeof the invention as set forth in the claims that follow. Thespecification and drawings are accordingly to be regarded in anillustrative rather than restrictive sense.

1. A rectifier, comprising: an AC input, including AC nodes; a DCoutput, including first and second DC nodes; a switching circuit coupledwith the AC input and with the DC output, the switching circuitincluding switching devices respectively configured to selectivelycouple one of the AC nodes with a respective one of the first and secondDC nodes according to a respective switching control signal; and acontroller configured to: compute a voltage reference according to: aload signal of the DC output, and a non-linear relationship between anelectrical load condition of the DC output and a DC bus voltage at theDC output, and generate the switching control signals according to thevoltage reference to cause the switching circuit to convert AC inputpower from the AC input to control the DC bus voltage at the DCoutput_(.;) wherein the non-linear relationship is a multi-segmentfunction with multiple segments having different respective slopes thatrelate the load condition of the DC output to the DC bus voltage at theDC output.
 2. (canceled)
 3. The rectifier of claim 1, wherein themulti-segment function includes more than two segments having differentrespective slopes that relate the load condition of the DC output to theDC bus voltage at the DC output.
 4. The rectifier of claim 1, whereinthe multi-segment function is piecewise linear.
 5. The rectifier ofclaim 1, wherein the multi-segment function includes a curvilinearsegment.
 6. A rectifier, comprising: an AC input, including AC nodes; aDC output, including first and second DC nodes; a switching circuitcoupled with the AC input and with the DC output, the switching circuitincluding switching devices respectively configured to selectivelycouple one of the AC nodes with a respective one of the first and secondDC nodes according to a respective switching control signal; and acontroller configured to: compute a voltage reference according to: aload signal of the DC output, and a non-linear relationship between anelectrical load condition of the DC output and a DC bus voltage at theDC output, and generate the switching control signals according to thevoltage reference to cause the switching circuit to convert AC inputpower from the AC input to control the DC bus voltage at the DC output;wherein the non-linear relationship is a second or higher order formulathat relates the load condition of the DC output to the DC bus voltageat the DC output.
 7. The rectifier of claim 6, wherein the controllerincludes a lookup table with entries that relate load conditions of theDC output to respective DC bus voltages at the DC output according tothe second or higher order formula.
 8. The rectifier of claim 6, whereinthe controller is configured to solve the second or higher order formulato compute the voltage reference for a measured load condition of the DCoutput for a given control cycle.
 9. The rectifier of claim 8, whereinthe controller is configured to: sample a root-mean-square line to linevoltage of the AC input for a given control cycle; compute a minimum DCbus voltage value according to the root-mean-square line to line voltagefor the given control cycle; compute a no load DC bus voltage valueaccording to the minimum DC bus voltage value for the given controlcycle; and compute the voltage reference for the given control cycleaccording to the load signal, non-linear relationship, and the no loadDC bus voltage value.
 10. The rectifier of claim 6, wherein thecontroller is configured to: sample a root-mean-square line to linevoltage of the AC input for a given control cycle; compute a minimum DCbus voltage value according to the root-mean-square line to line voltagefor the given control cycle; compute a no load DC bus voltage valueaccording to the minimum DC bus voltage value for the given controlcycle; and compute the voltage reference for the given control cycleaccording to the load signal, non-linear relationship, and the no loadDC bus voltage value.
 11. The rectifier of claim 1, wherein the loadsignal of the DC output is a DC output current signal of the DC output.12. A power conversion system, comprising a first rectifier, comprising:a first AC input, including first AC nodes, a first DC output, includingfirst and second DC nodes, a first switching circuit coupled with thefirst AC input and with the first DC output, the first switching circuitincluding first switching devices respectively configured to selectivelycouple one of the first AC nodes with a respective one of the first andsecond DC nodes according to a respective first switching controlsignal, and a first controller configured to: compute a first voltagereference according to: a first load signal of the first DC output, anda first non-linear relationship between a first electrical loadcondition of the first DC output and a DC bus voltage across the firstand second DC nodes, and generate the first switching control signalsaccording to the first voltage reference to cause the first switchingcircuit to convert AC input power from the first AC input to control theDC bus voltage; and a second rectifier, comprising: a second AC input,including second AC nodes, a second DC output connected to the first andsecond DC nodes of the first DC output, a second switching circuitcoupled with the second AC input and with the first DC output, thesecond switching circuit including second switching devices respectivelyconfigured to selectively couple one of the second AC nodes with arespective one of the first and second DC nodes according to arespective second switching control signal, and a second controllerconfigured to: compute a second voltage reference according to: a secondload signal of the second DC output, and a second non-linearrelationship between a second electrical load condition of the second DCoutput and the DC bus voltage, and generate the second switching controlsignals according to the second voltage reference to cause the secondswitching circuit to convert AC input power from the second AC input tocontrol the DC bus voltage.
 13. The power conversion system of claim 12,wherein the respective first and second non-linear relationships aremulti-segment functions with multiple segments having differentrespective slopes that relate the load condition of the DC output to theDC bus voltage.
 14. The power conversion system of claim 12, wherein therespective first and second non-linear relationships are second orhigher order formulas that relate the respective load conditions of thefirst and second DC outputs to the DC bus voltage.
 15. The powerconversion system of claim 12, wherein the first load signal of thefirst DC output is a first DC output current signal of the first DCoutput; and wherein the second load signal of the second DC output is asecond DC output current signal of the second DC output.
 16. A method,comprising: computing a voltage reference according to: a load signal ofa DC output of an active rectifier, and a non-linear relationshipbetween an electrical load condition of the DC output and a DC busvoltage at the DC output; and generating switching control signals forthe active rectifier according to the voltage reference to cause aswitching circuit of the active rectifier to convert AC input power froman AC input to control the DC bus voltage; wherein the non-linearrelationship is a multi-segment function with multiple segments havingdifferent respective slopes that relate the load condition of the DCoutput to the DC bus voltage at the DC output.
 17. The method of claim16, further comprising: sampling a root-mean-square line to line voltageof the AC input for a given control cycle; computing a minimum DC busvoltage value according to the root-mean-square line to line voltage forthe given control cycle; computing a no load DC bus voltage valueaccording to the minimum DC bus voltage value for the given controlcycle; and computing the voltage reference for the given control cycleaccording to: the load signal, non-linear relationship, and the no loadDC bus voltage value.
 18. (canceled)
 19. A method, comprising: computinga voltage reference according to: a load signal of a DC output of anactive rectifier, and a non-linear relationship between an electricalload condition of the DC output and a DC bus voltage at the DC output;and generating switching control signals for the active rectifieraccording to the voltage reference to cause a switching circuit of theactive rectifier to convert AC input power from an AC input to controlthe DC bus voltage; wherein the non-linear relationship is a second orhigher order formula that relates the load condition of the DC output tothe DC bus voltage at the DC output.
 20. The method of claim 16, whereinthe load signal of the DC output is a DC output current signal of the DCoutput.
 21. The rectifier of claim 6, wherein the load signal of the DCoutput is a DC output current signal of the DC output.
 22. The method ofclaim 19, wherein the load signal of the DC output is a DC outputcurrent signal of the DC output.